quantitative forecast of when each species is likely to be extirpated
from each geographical location.
Despite the limitations of TPCs in accounting for temporal car-
ryover and dynamical effects, the lack of obvious alternatives calls for
strategies to make these approaches more robust to real-world condi-
tions38, such as by integrating more realistic, fine-scaled temperature
variation into our predictive models than previous studies. Although
bioclimate envelope approaches have been criticized for not account-
ing for important ecological factors, such as species interactions and
dispersal, when attempting to predict the ecological effects of climate
change39–42, we have shown that even under ideal conditions when the
influence of such factors can be assumed to be negligible, statistical
frameworks that ignore the dynamical consequences of temperature
variation are likely to yield inaccurate forecasts of the impact of climate
change on organisms. Our results show that accounting for shifts in the
entire statistical distribution of temperature over time via dynamical
models can better capture the cumulative effects of climate-mediated
changes in thermal stress on extinction risk.
By bringing together climate data and a minimal dynamical model
from ecology, we demonstrated a strong and systematic amplifica-
tion of extinction risk in ectotherms due to projected changes in
fine-grained temperature variability. Furthermore, our finding of
greater risk to sub-tropical than tropical species highlights the impor-
tance of accounting for the dynamical effects of projected changes in
the mean as well as the variance of temperature over the course of the
twenty-first century to accurately predict the response of ecological
systems around the globe.
Online content
Any methods, additional references, Nature Research reporting sum-
maries, source data, extended data, supplementary information,
acknowledgements, peer review information; details of author contri-
butions and competing interests; and statements of data and code avail-
ability are available at https://doi.org/10.1038/s41558-022-01490-7.
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Articlehttps://doi.org/10.1038/s41558-022-01490-7
Methods
CMIP6 simulations
We obtained CMIP6 climate simulations for the historical forcing
period (1850–2014) and future emissions scenario SSP5-8.5 (2015–
2100) via the CMIP6 data portal (https://esgf-node.llnl.gov/search/
cmip6/). Eight models from CMIP6 (AWI-CM-1-1-MR, BCC-CSM2-MR,
CESM2, EC-Earth3, INM-CM5-0, MPI-ESM1-2-HR, MRI-ESM2-0 and
NorESM2-MM) were selected on the basis of availability of daily air
temperature at surface (‘tas’) at a 100 km nominal resolution at the
time of download. While ‘tas’ at sub-daily frequencies is available
for some models, daily data was selected to maximize the ensemble
size. We resampled all datasets to a common 1° by 1° grid spanning
−90° to 90° latitude and 0° to 360° longitude, and to a standard cal-
endar without leap years. Spatial regions were defined on the basis
of latitude as Northern Hemisphere Extra-tropics (90° S to 30° S),
Tropics (30° S to 30° N) and Southern Hemisphere Extra-tropics
(30° N to 90° N).
Statistical analyses of climate data
Quantile regression. Trends in the percentile values of global and
regional temperature distributions were computed via quantile
regression. Quantile regression can comprehensively model hetero-
geneous conditional distributions, where the relationship between
the quantiles of the dependent variable and the independent variable
is different from the relationship between the means of the depend-
ent variable and the independent variable. We applied quantile
regression to analyse trends with respect to time at various percentile
values (P2.5, P10, P20, P30, P40, P50, P60, P70, P80, P90, P97.5). Analyses were
performed using the R package quantreg, with significance level
α = 0.1 and the default Barrodale and Roberts method to return
confidence intervals for the estimated parameters. To obtain the
ensemble mean trends, we calculated the mean slope, upper bound
and lower bound across the eight climate models at each geographi-
cal location, then computed spatial averages for the full globe and
three latitudinal regions.
Variance. Trends in the magnitude of temporal variation of air tem-
perature were examined at each geographical location using a mov-
ing window approach. First, temperature was detrended by fitting a
piecewise linear regression against time with Python package pwlf
at each geographical location and extracting the residuals. Then, the
temperature time series were divided into 10 yr windows starting in
years 1855 through 2085 so as not to combine historical and future
simulations (pre- and post-01-01-2015), and the variance of daily air
temperature was calculated for each window. Windows were selected
with no overlap to avoid statistical issues due to non-independence of
estimates taken from partially overlapping time windows20.
Scale-specific variability. Scale-specific variability was quantified
using time-frequency decomposition. Specifically, at each geographi-
cal location, wavelet analysis was conducted on multi-model mean
temperature using the R package biwavelet43. Wavelet analysis resolves
both the time and frequency domains of a signal (here a time series) via
the wavelet transform. This is achieved via the convolution of a mother
wavelet function and a time series across a set of windows τ and scales s.
We chose to use the Morlet wavelet, which represents a sine wave
modulated by a Gaussian function:44
ψ0 (t) = π−1/4eiωt
0 e−r2/2
where i is the imaginary unit, t represents non-dimensional time, and
ω0 = 6 is the non-dimensional frequency3. The continuous wavelet
transform of a discrete time series x (t) with equal spacing δt and length
T is defined as the convolution of x (t) with a normalized Morlet
wavelet:44,45
Nature Climate Change
Wx (s, τ) =
√√√
√
δt
s
T−1
∑
t=0
x (t) ψ0 ∗ (
(t − τ) δt
s
)
where * indicates the complex conjugate. By varying the wavelet scale
s (that is, dilating and contracting the wavelet) and translating along
localized time position τ, one can calculate the wavelet coefficients
Wx (s, τ) across the different scales s and positions τ. These wavelet coef-
ficients can be used to compute the bias-corrected local wavelet power,
which describes how the contribution of each frequency or period in
the time series varies over time:44,46,47
W2
x (s, τ) = 2s |Wx (s, τ)|
2where 2s is the bias correction factor46. The
scale s of the Morlet wavelet is related to the Fourier frequency f:47,48
1
f
=
4πs
ω0+√2+ω2
0
.
When ω0 = 6, the scale s is approximately equal to the reciprocal
of the Fourier frequency f, so period p ≈ s. The local wavelet power
spectrum can then be visualized via heat maps and contour plots45,47.
From the resulting local wavelet power spectrum heat map with time
on the x axis, period (scale) on the y axis and power on the z axis,
scale-averaged wavelet power was computed at annual (between 3 d
and 2 yr) and multi-annual (between 2 yr and 30 yr) periodicities. This
was achieved by taking the weighted sum of the local wavelet power
across all scales for each time location τ:44,47
W−2
x
(τ) =
δjδt
Cδ
J
∑
j=0
2
||Wx (sj, τ)||
sj
where Cδ = 0.776 for the Morlet wavelet, δJ represents the spacing
between successive scales and δt represents the spacing between suc-
cessive time locations44. Scale-averaged power was then regressed
against time using Generalized Least Squares (GLS) regression for the
period 1850–2100 at each geographic location. To determine the
robustness of results to the choice of period for scale averaging, we
also performed analysis of trends separately at interannual (between
2 yr and 7 yr) and multi-annual (between 7 yr and 30 yr) scales and found
qualitatively similar results.
Temporal autocorrelation. The temporal autocorrelation of air
temperature was quantified by calculating the spectral exponent at
each geographical location20. As described above, temperature was
detrended by fitting a piecewise linear regression at each geographical
location and extracting the residuals. The detrended temperature was
divided into 10 yr windows starting in years 1855 through 2085. Fourier
transforms of each time series were computed via fast Fourier trans-
form using the Python package NumPy. Periodograms were prepared
with frequency on the x axis and power spectral density on the y axis.
The spectral exponent, β, was calculated as the slope of the regression
line relating log-transformed power to log-transformed frequency.
β expresses the relative contributions of frequencies to the power
spectrum. In the case of equal contribution from all frequencies, β = 0.
Greater contribution from low frequencies than from high frequencies
results in a more negative value of β and indicates greater temporal
autocorrelation in the time domain.
Analysis of decadal trends. For each climate model, GLS regression
was used to detect statistically significant trends (P < 0.05) in variance
and temporal autocorrelation with respect to time in the presence of
potentially autocorrelated residuals. To measure inter-model agree-
ment, we calculated the multi-model mean trend as the mean of trends
calculated for each of the 8 models at each geographic location, then
assessed the proportion of models that agreed with the sign of the
multi-model mean trend. Inter-model agreement was considered as
statistically significant at the α = 0.1 level on the basis of a binomial
test. ANCOVA was used to quantify the relationship between temporal
Articlehttps://doi.org/10.1038/s41558-022-01490-7
autocorrelation and time while accounting for potential differences
between land and sea environments. Statistically significant main
effects and interactions were reported for P < 0.05.
dynamics. Specifically, we used the r − α logistic growth model to
simulate temperature-dependent growth and negative
density-dependence:
Modelling temperature impacts on ecology
Thermal tolerance data. We obtained experimentally derived ther-
mal tolerance parameters for a set of terrestrial ectotherms (n = 38)
published by Deutsch et al.5 and used them to predict physiological
response to CMIP6-simulated temperature. Deutsch et al.5 gathered
data from 31 thermal performance studies published between 1974
and 2003 based on a collection of insects from 35 different locations.
For each species, experimental intrinsic growth rates at multiple tem-
peratures were used to fit a TPC, yielding least-squares estimates of
key parameters such as critical thermal maximum (CTmax), optimum
temperature (Topt), and sigma (σ). We used a numerical scheme to
reconstruct the curves whereby the rise in performance up to Topt was
modelled as Gaussian and the decline beyond Topt was quadratic5,49:
P (T) =
⎧
⎪
⎨
⎪
⎩
exp [− (
T−Topt
2σ
2
)
] for T ≤ Topt
1 − (
T−Topt
Topt−CTmax
2
)
for T > Topt
.
(1)
This allowed negative growth rates to arise at high temperatures,
but growth rates were bound at zero at low temperatures. Negative
performance values indicate that mortality surpasses reproduction
rates. Because P(T) is capped at 1 under this numerical scheme, P(T)
represents the relative fitness of each species based on its normalized
maximum growth rate. However, scaling this relative or normalized
maximum growth rate by two orders of magnitude (that is, by a factor
of 0.1 or 10.0) had limited quantitative and no qualitative impact on
our results (Extended Data Fig. 4). Overall, increasing the growth rate
scaling factor had no impact on population stability but promoted
extinction risk.
Isolation of temperature aspects. To isolate projected changes in
mean temperature and variability, we transformed the future (2050–
2100) time series using z-score normalization. Using this approach, we
modified projected time series to match the historical (1950–2000)
mean and/or standard deviation. Working in 10 yr moving windows
between 2050 and 2100, each series xi with mean m1 and standard
deviation s1 was transformed to series yi with mean m2 and standard
deviation s2:
yi = m2 + (xi − m1)
s2
s1
.
(2)
According to the scenario, m2 and s2 were alternatively defined
as (1) high emissions scenario mean and standard deviation (‘Mean,
variance and autocorrelation’), (2) high emissions scenario mean and
historical standard deviation (‘Mean and autocorrelation’), (3) histori-
cal mean and high emissions scenario standard deviation (‘Variance
and autocorrelation’) and (4) historical mean and standard deviation
(‘Autocorrelation’). High emissions scenario statistics refer to the
properties of future series xi and confer no change to that aspect of
the time series.
Population dynamical modelling. To model the effects of tempera-
ture change on the stability and extinction probability of global ecto-
therm populations, temperature dependence was integrated in the
growth rate term of a population dynamical model50. While more com-
plex synecological models can capture a range of community-level
effects including competition and predation, we chose to model first
order autecological dynamics to produce foundational insights about
the role of temperature fluctuations on single-species population
Nature Climate Change
dN
dt
= N (rt − αN)
(3)
with population size N, time t, temperature-dependent growth rate rt,
and self-regulation in the form of intraspecific competition α. This r − α
logistic model is easily interconvertible with the classical r − K formula-
tion (r/α = K), but has the advantage of handling negative values of r
without issues51. This approach is sensitive to the effects of tempera-
tures at and above the critical thermal maximum, which can yield
negative growth rates that are important for determining population
dynamics as well as long-term fitness.
We extracted time series of daily temperature at the source loca-
tions for each species from the ensemble of eight climate simulations.
Daily intrinsic growth rates were computed from temperature using
equation (1), incorporated into the r − α logistic growth model depicted
in equation (3), and the model was then numerically solved using the
explicit Runge-Kutta method of order 5(4) implemented in the Python
SciPy package to obtain daily population densities. Rather than delin-
eating active periods, which may shift under climate change, we con-
sidered the full year to account for potential changes in fitness due to
shifts in activity.
The sensitivity of the results to strong (α = 1) and weak (α = 0.1)
self-regulation was examined and found to be extremely limited
(Extended Data Fig. 5). We also assessed the sensitivity of our results to
absolute rather than relative or normalized growth rates by scaling rt by
a factor of 0.1 or 10 in our simulations. Scaling rt by two orders of mag-
nitude in this manner had very little quantitative and no qualitive impact
on our results. This suggests that the effects of temperature fluctuations
on changes in the spatiotemporal distribution of population abundance,
stability and extinction were not contingent upon the use of relative
fitness (that is, normalized growth rate) versus absolute fitness (that is,
growth rate scaled by a factor of 0.1 or 10). These sensitivity analyses also
showed that our results are robust to temperature-mediated changes in
the maximum instantaneous growth rate52,53.
Analysis of population changes. To quantify temperature-driven
changes in ecological stability and extinction probability, we compared
population sizes and dynamics between a historical period (1950–
2000) and a future period (2050–2100). Here we defined latitudinal
regions according to traditional delineations in ecology: Northern
Hemisphere Extra-tropics, 60° S to 23° S; Tropics, 23° S to 23° N; and
Southern Hemisphere Extra-tropics, 23° N to 60° N.
Population abundance was computed as the mean population
size (N) for a time period. Population stability was computed as the
inverse of the coefficient of variation, or mean population divided
by population standard deviation. Percent changes in population size
and stability were computed for each of the climate models as
(future − historical) /historical × 100% and plotted without outliers in
Fig. 4. Statistically significant changes in population abundance and
stability between the historical and future periods were identified via
the Mann–Whitney U-test, with the eight models as replicates.
Extinction probability was quantified as the proportion of ensem-
ble simulations for which the population declined to zero during a
50 yr simulation. Changes in extinction probability were calculated as
the difference between future and historical extinction probabilities.
Statistically significant changes in extinction probability were identi-
fied on a regional basis via the Mann–Whitney U-test.
Reporting summary
Further information on research design is available in the Nature
Research Reporting Summary linked to this article.
Articlehttps://doi.org/10.1038/s41558-022-01490-7
Data availability
The CMIP6 simulation data used in this paper are available via the data
portal https://esgf-node.llnl.gov/search/cmip6/. The ecology data are
available for download at https://doi.org/10.1073/pnas.0709472105.
Code availability
The code can be accessed on GitHub at https://github.com/KateDuffy/
climate-change-ecology54.
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Acknowledgements
This work was primarily supported by the National Science
Foundation (NSF) grant CCF-1442728 while K.D. was a PhD
student at the SDS Lab in Northeastern University. Additional
support was provided for K.D. and A.R.G. by NSF SES-1735505
and for T.C.G. by NSF OCE-2048894. We acknowledge the
background support from a previous NSF Expeditions in
Computing grant (award no. 1029711) and an ongoing DOD
Strategic Environmental Research and Development Program
funding (no. RC20-1183). K.D. and A.R.G. acknowledge support
from the NASA Ames Research Center.
Author contributions
K.D., T.C.G. and A.R.G. conceived, designed and refined the project.
K.D. performed the data analysis and modelling. K.D., T.C.G. and A.R.G.
interpreted the results. K.D. wrote the manuscript with contributions
from T.C.G. and A.R.G.
Competing interests
The authors declare no competing interests.
Additional information
Extended data is available for this paper at
https://doi.org/10.1038/s41558-022-01490-7.
Supplementary information The online version
contains supplementary material available at
https://doi.org/10.1038/s41558-022-01490-7.
Correspondence and requests for materials should be addressed to
Kate Duffy.
Peer review information Nature Climate Change thanks David Vasseur
and the other, anonymous, reviewer(s) for their contribution to the
peer review of this work.
temperature interact to determine physiological tolerance and
fitness. Physiol. Biochem. Zool. 84, 543–552 (2011).
Reprints and permissions information is available at
www.nature.com/reprints.
Nature Climate Change
Articlehttps://doi.org/10.1038/s41558-022-01490-7
Extended Data Fig. 1 | Temperature variation at multiple timescales
contributes to trends in overall variance. a-b Temporal trends in the power
of variation at sub-annual to annual periodicities (3-days to 2-years) (a) and
multi-annual periodicities (2-30 years) (b). Trends represent the slope obtained
by regressing wavelet power at each geographical location against time.
Countervailing trends are found in the Arctic, where the power of short term,
high-frequency fluctuations is decreasing and the power of more persistent,
low-frequency fluctuations is increasing.
Nature Climate Change
Articlehttps://doi.org/10.1038/s41558-022-01490-7
Extended Data Fig. 2 | Drivers of changes in stability (analysis includes
both pre- and post-extinction period). Kernel density plots illustrate the
relationships between population mean and population standard deviation
in the historical period and the future climate change period. The grey 1:1 line
divides the more stable regime (high-mean/low-variance; below line), and the
less stable regime (low-mean/high-variance; above line). Bimodal distributions
emerge in the extra-tropics, with some species at low abundance and standard
deviation, and a larger cluster of species at high abundance and standard
deviation.
Nature Climate Change
Articlehttps://doi.org/10.1038/s41558-022-01490-7
Extended Data Fig. 3 | Drivers of changes in stability (analysis only includes pre-extinction period). When only pre-extinction dynamics are analyzed, significant
changes in population abundance persist in the extra-tropics; changes in population standard deviation become significant for NHEX and TROP and remain non-
significant for SHEX.
Nature Climate Change
Articlehttps://doi.org/10.1038/s41558-022-01490-7
Extended Data Fig. 4 | Scaling of the intrinsic growth rate has moderate
effects on temperature-driven impacts on population stability and
extinction risk. Results exhibited limited sensitivity to the choice of smaller
(scaling factor = 0.1; a,b) and larger (scaling factor = 10.0; e,f) intrinsic growth
rates. Although larger growth rates were more strongly associated with deceased
stability and increased extinction risk than smaller growth rates, the latitudinal
patterns and effect sizes were consistent with the changes in population stability,
c, and extinction probability, d, observed under normalized growth rates.
Nature Climate Change
Articlehttps://doi.org/10.1038/s41558-022-01490-7
Extended Data Fig. 5 | Temperature-driven effects on population stability and extinction risk are robust to the degree of population self-regulation. Results
exhibited limited sensitivity to strong (α = 1; Fig. 4) and weak (α = 0.1; above) self-regulation in the form of crowding effects. Latitudinal patterns and effect sizes were
consistent for changes in population stability, a, and extinction probability, b.
Nature Climate Change
Articlehttps://doi.org/10.1038/s41558-022-01490-7
Extended Data Fig. 6 | Patterns of species risk are sensitive to treatment of
high-temperature performance. Change in the average fitness (growth rate)
driven by daily temperature between the historical period (1950-2000) and
future period (2050-2100) when allowing negative growth rates above CTmax
as in Vasseur et al. 2014 (a) and when performance values are bounded by 0
above CTmax as in Deutsch et al. 2008 (b). Change in the average fitness (growth
rate) driven by monthly mean temperature between the historical period and
future period when allowing negative growth rates above CTmax (c) and when
performance values are bounded by 0 above CTmax (d).
Nature Climate Change
Articlehttps://doi.org/10.1038/s41558-022-01490-7
Extended Data Fig. 7 | Increased stability is negatively related to extinction
probability. Regression relationships in our simulations are presented a, when
considering only the pre-extinction time period and b, when taking into account
the full 50-year periods. Regardless of largely positive (b) or mixed (a) changes in
stability, there is generally a weak but significant negative relationship between
stability and extinction probability globally (p-value < 0.05).
Nature Climate Change
Articlehttps://doi.org/10.1038/s41558-022-01490-7
Corresponding author(s):
Kate Duffy
Last updated by author(s): Aug 23, 2022
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